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Approximate ternary Jordan derivations on Banach ternary algebras. (English) Zbl 1214.46034

Summary: Let \(A\) be a Banach ternary algebra over a scalar field \(\mathbb R\) or \(\mathbb C\) and \(X\) be a ternary Banach \(A\)-module. A linear mapping \(D: (A,[\,]_ A)\to(X,[\,]_ X)\) is called a ternary Jordan derivation if \(D([xxx]_ A)=[D(x)xx]_ X+[xD(x)x]_ X+[xxD(x)]_ X\) for all \(x\in A\). In this paper, we investigate ternary Jordan derivations on Banach ternary algebras, associated with the following functional equation \(f((x+y+z)/4)+f((3x-y-4z)/4)+f((4x+3z)/4)=2f(x)\). Moreover, we prove the generalized Ulam-Hyers stability of ternary Jordan derivations on Banach ternary algebras.
In a comment C.-G. Park and M. Eshaghi Gordji [J. Math. Phys. 51, No. 4, 044102 (2010)] state: The mapping \(f\) in Lemma 2.2 of B. Savadkouhi et al. is identically zero and all of the results are trivial. In this note, we correct the statements of the results and the proofs.

MSC:

46K05 General theory of topological algebras with involution
39B82 Stability, separation, extension, and related topics for functional equations
47B47 Commutators, derivations, elementary operators, etc.
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